Abstract

The system of Gegenbauer or ultraspherical polynomials is a classical family of polynomials orthogonal with respect to the weight function on the support interval Integral functionals of Gegenbauer polynomials with integrand where is an arbitrary function which does not depend on or λ, are considered in this paper. First, a general recursion formula for these functionals is obtained. Then, the explicit expression for some specific functionals of this type is found in a closed and compact form; namely, for the functionals with equal to and which appear in numerous physico-mathematical problems. Finally, these functionals are used in the explicit evaluation of the momentum expectation values and of the -dimensional hydrogenic atom with nuclear charge The power expectation values are given by means of a terminating hypergeometric function with unit argument, which is a considerable improvement with respect to Hey’s expression (the only one existing up to now) which requires a double sum.